Mathematical Sciences: Harmonic Analysis on Homogeneous Spaces, C*-Algebraic Index Theory

9625336利普斯曼教授和罗森博格教授计划继续他们在群表示理论、齐次空间分析和C*-代数指数理论方面的工作。利普斯曼教授将研究齐次空间的傅立叶分析，它与不变微分算子和各种轨道结构的关系，以及分布Frobenius互易的概念。Rosenberg教授将研究正数量曲率流形，类流形空间上的指数理论，以及算子代数的K-理论和L-理论。利普斯曼和罗森伯格教授将共同研究关于李群的Paley-Wiener定理和不确定性原理。这个项目的重点将放在分析(例如，研究偏微分方程解)和广义解释的几何结构之间的相互作用上。分析与几何之间的关系是现代数学和物理学中最重要的主题之一。利普斯曼教授和罗森博格教授还将共同努力，通过用计算机项目加强常见的多变量微积分课程，帮助向本科生传达这一主题，强调基本的几何和物理原理。***

9625336 Lipsman Professors Lipsman and Rosenberg plan to continue their respective work in group representation theory, analysis on homogeneous spaces, and C*-algebraic index theory. Professor Lipsman will study the Fourier analysis of homogeneous spaces, its relation to invariant differential operators and various orbital constructs, and the notion of distributional Frobenius reciprocity. Professor Rosenberg will study manifolds of positive scalar curvature, index theory on manifold-like spaces, and K-theory and L-theory of operator algebras. Professors Lipsman and Rosenberg will jointly study Paley-Wiener Theorems and uncertainty principles on Lie groups. The emphasis of this project will be on the interplay between analysis (for example, study of the solutions of partial differential equations) and geometrical structure, broadly interpreted. The relationship between analysis and geometry is one of the most important themes in modern mathematics and physics. Professors Lipsman and Rosenberg will also jointly work to help convey this theme to undergraduates by enhancing the usual multivariable calculus course with computer projects emphasizing the underlying geometrical and physical principles. ***